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Eng. Fracture Mechanics, vol. 70, 2003, 233-248

THE CALCULATION OF ADHESIVE FRACTURE ENERGIES IN MODE I:

REVISITING THE TAPERED DOUBLE CANTILEVER BEAM (TDCB) TEST.

B.R.K. BLACKMAN †, H. HADAVINIA, A.J. KINLOCH, M. PARASCHI

AND J.G. WILLIAMS

Department of Mechanical Engineering, Imperial College of Science, Technology and

Medicine, Exhibition Road, London SW7 2BX. UK.

Abstract—Analytical corrections have been derived for a beam theory

analysis for the adhesively-bonded tapered double cantilever beam (TDCB)

test specimen to account for the effects of beam root rotation and for the real,

as opposed to idealised, profile of the beam as required experimentally. A

number of adhesive-substrate combinations were tested according to a new

test protocol and the new analysis method for data reduction is compared

critically with the existing simple beam theory and experimental compliance

approaches. Correcting the beam theory for root rotation effects is shown to

be more important than correcting only for the effects of shear deformation of

the substrates. Results from a finite element analysis, using a cohesive zone

model, also showed close agreement with the proposed new corrected beam

theory (CBT) analysis method.

Keywords—adhesive joints, tapered double cantilever beam (TDCB),

fracture mechanics, simple beam theory (SBT), corrected beam theory (CBT),

root rotation, cohesive zone.

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† Corresponding author, email: b.blackman@ic.ac.uk, fax: +44 (0)20 7594 7017

NOMENCLATURE

A1,2 integration constants

a crack length

aint intercept on the crack length axis of a C versus a plot

B1,2 integration constants

b width of test specimen

C compliance of the beam

E Young’s modulus of the substrate

Ea Young’s modulus of the adhesive

GIC the adhesive fracture energy

h height of the beam

ha thickness of adhesive layer

ho height of initial, non-profiled section of the TDCB

k the beam foundation stiffness

m specimen geometry factor

m̂ a modified specimen geometry factor

P load applied to the test specimen

R ratio of E/Ea

uo the load-line displacement of both arms of the specimen during a test

v(x) vertical displacement of the beam at distance x from load-line.

x distance along the beam measured from the load-line

xo length of straight section of the beam measured from the load-line

δc a critical separation used in the cohesive zone model

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∆ beam root rotation correction

λ ratio of 2h/ha

σmax stress parameter used in the cohesive zone model

CBT corrected beam theory

CZM cohesive zone model

DCB double cantilever beam

ECM experimental compliance method

FE Finite element

SBT simple shear-corrected beam theory

TDCB tapered double cantilever beam

1. INTRODUCTION

The application of fracture mechanics to adhesive joints dates back to the 1960s when

Ripling et al [1] and later Mostovoy et al [2] studied experimental methods to determine the

plane-strain fracture toughness of bonded metallic joints. This work led to an ASTM standard

[3] which used a simple shear-corrected beam theory to deduce the values of GIC from either

adhesively bonded double cantilever beam (DCB) or tapered double cantilever beam (TDCB)

test specimens. However, this analysis did not consider the effects of beam root rotation, nor

did it account for the real, as opposed to idealized, profile of the tapered beam. Indeed,

Mostovoy and co-workers [2] noted that the simple shear-corrected beam theory required a

correction for rotations at the assumed ‘built-in’ end of the beam and determined an empirical

rotation correction, ao, which could be treated as an increase in the crack length. They found

experimentally that the correction was approximately equal to 0.6h, where h was the height of

the beam for the DCB test specimen. However, this correction was not implemented in the

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ASTM standard [3], nor was it suggested how the correction could be used with the height

tapered beams employed in the TDCB test specimen.

In the present work, an analytical correction for beam root rotation is derived for use

with the tapered double cantilever beam specimen. In addition, the beam theory analysis

described considers the actual profile of the test specimen that includes an initial linear

section (where the beam height is a constant) as shown in Figure 1. Experimental results

obtained from this new, corrected analysis are compared to both the simple shear-corrected

beam analysis employed in [3] and also with an experimental compliance calibration method.

Results for two different adhesives and two different beam substrate materials, i.e. aluminium

alloy and mild steel, are presented. Further, the analytical and experimental results are

compared with a finite element analysis employing a cohesive zone model. Finally, the new

analysis scheme proposed for the TDCB test has been incorporated into a new test protocol

that has been critically assessed in a round-robin programme organized by the European

Structural Integrity Society (ESIS), under the Technical Committee on Polymers, Adhesives

and Composites. The protocol and initial round-robin results can be found in [4] and a full

presentation of the round-robin results will be published shortly [5].

2. ANALYTICAL STUDIES

The adhesive fracture energy, GIC, for the tapered double cantilever beam adhesive

joint specimen may be determined directly using the Irwin-Kies equation, (1):

dadC

bPGIC 2

2

= (1)

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where P is the applied load, b the specimen width, C the compliance of the substrate beam

(given by the displacement divided by the load) and a is the crack length. The value of dC/da

can be measured experimentally and thus the GIC value calculated. In the present work,

equation (1) is described as the experimental compliance method (ECM) of analysis. The

analyses that follow all aim to solve for dC/da.

2.1. Simple shear-corrected beam theory (SBT) analysis

Mostovoy et al [2] were the first to propose the use of the height tapered double

cantilever beam, (TDCB), test geometry for the purpose of measuring the resistance to crack

growth in adhesive joints. By first considering a double cantilever beam specimen, the

compliance of the beam was determined by considering the contributions from bending and

shear deflections. This analysis led to equation (2):

+=

hha

EbdadC 138

3

2

(2)

where E and h are the substrate modulus and height respectively. They proposed that if the

height of the beam was carefully profiled, then dC/da could be held constant by ensuring that

the quantity in brackets in equation (2) was also a constant. Thus, if the beam was machined

to a geometry factor, m, given by:

constanthh

am =+=13

3

2

(3)

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then the value of dC/da would be constant, i.e. the compliance would change linearly with

crack length and also, by substituting into equations (2) and (1) the adhesive fracture energy

was expressed as:

mEbP

hha

EbPGIC ⋅=

+⋅= 2

2

3

2

2

2 4134 (4)

Indeed, equation (4) was applied to either straight, double cantilever beam specimens or

height tapered beams as outlined in the ASTM standard [3]. Such specimens have become

very popular and are widely used for both static and fatigue testing and also for the ageing of

adhesive joints in various environments.

2.2. Corrected beam theory (CBT) analyses

2.2.1. Introduction

As stated previously, the above analysis considers the deflections of the substrate

beams due to bending and shear but does not allow for the important contribution to the

compliance from the deflection and rotation at the beam root, i.e. at the assumed built-in crack

tip. Indeed, in the present work it is shown that these effects are more important than those

associated with shear deformations when predicting the compliance and GIC values of TDCB

specimens manufactured with metallic substrates. Equation (3), which defines the geometry

factor of the tapered double cantilever beam, is only weakly dependent upon the shear term

1/h. In the analysis that follows, equation (3) has been simplified to equation (5):

3

23ham = (5)

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i.e. the shear-correction term 1/h has been neglected in the calculation of m. This

simplification allows the bending equations to be readily integrated and enables a simple

scheme to be proposed to account for root rotation. For the beams used in the present work,

the geometry factor, as defined by equation (3), was always equal to 2mm-1. Thus, the error

in m imposed by the above simplification was –2% for a crack length of 100mm and –1.3%

for a crack length of 200mm, which represents the typical extent of crack propagation

observed in the experiments reported here.

2.2.2. Determining the compliance of the TDCB

To enable the TDCB specimen to be readily tested, the initial section of the beam is

not profiled, thus enabling holes to be drilled through the metallic substrates as shown in

Figure 1. Loading pins may then be inserted through the holes and the beam loaded in mode I

tension. Beams are manufactured such that the geometry factor m, given by equation (3), is a

constant. A single arm of the tapered beam is shown in Figure 2 with notation shown. In the

present analysis, the height of the straight section of the beam is termed ho and this extends a

distance xo from the load-line. The height of the profiled section of the beam is given by h

which is a function of the distance x, again measured from the load-line. The beam has a

width b and a load P and displacement uo/2, is applied at x=0 during the test. Because of the

discontinuity of the beam profile at x=xo, beam theory has been applied to the two sections of

the beam separately. Firstly, the straight section of the beam where 0<x<xo and secondly the

profiled section where x>xo. If the vertical displacement along the beam is given by v(x),

then:

(i) For the straight beam section, 0<x<xo

From beam theory we can write:

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+= 1

2

2 24 APx

Ebxm

dxdv

o

(6 )

and

++= 11

3

2 64 BxAPx

Ebxmv

o

(7)

where A1 and B1 are integration constants. The boundary condition is that at x=0, v=uo/2

where uo is the total displacement of two arms of the TDCB, i.e. the measured displacement

during a test. Therefore, from (7):

12

42

BEbx

mu

o

o ⋅=

(ii) For the profiled beam section, x>xo

( )2ln4 AxPEbm

dxdv

+= (8)

and

( )( )( )221ln4 BxAxPxEbmv ++−= (9)

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where A2 and B2 are two further integration constants. The boundary conditions are now the

usual built-in beam assumptions, i.e. when x=a, v=dv/dx=0, where a is the crack length. This

leads to:

A2 = -Plna and B2 = Pa.

Now, equating the expressions for v and dv/dx at x=xo allows the constants A1 and A2 to be

determined and thus equation (9) becomes:

−⋅== o

o xaPEbmuv

324

2

and as we define compliance as uo/P, then for x>xo we can write:

−= oxa

EbmC

328 (10)

Equation (10) implies that the relationship between compliance and crack length is

linear for x>xo, as expected, but does not pass through the origin. Indeed, when C=0,

a=(2/3)xo which means that, for a typical beam with xo=50mm, a positive C(a) intercept of

33.3mm would be predicted by this bending analysis. The expected variation in compliance

with distance x is depicted in Figure 3. For x<xo, the compliance is proportional to x3, but for

x>xo, compliance is directly proportional to x. Differentiating equation (10) recovers the

ASTM equation (2), so whilst identifying the C(0) intercept, equation (10) does not alter the

calculation of GIC. Equation (10) will be described here as the full-profile SBT analysis

method to emphasise that, in its derivation, the full beam profile has been considered.

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However, it is known that the built-in beam assumptions used in this analysis lead to errors in

compliance and hence errors in GIC. These are now considered.

2.2.3. Correcting the compliance for beam root rotation

Various schemes have been reported in the literature for determining the root rotation

correction for beam specimens. For example, Kanninen modelled the DCB specimen as a

beam on an elastic foundation [6]. Each arm was modelled as a cantilever beam supported

on a foundation of stiffness k per unit length. Solving the governing differential equation led

to a correction for beam root rotation via ∆. In effect, the crack length, a, was replaced by

a+∆ , where

k

Ebh3

34 =∆ (11)

where h and b were the height and width respectively of the DCB arm. The foundation

stiffness k was deduced by considering the elastic stretching of the foundation, i.e. the beam

material. This led to:

heih 64.0..61 4

1

≈

= ∆∆ (12)

In a later analysis which included the shear deformation of the beam, Kanninen [7]

showed that the effects of both shear deflection and beam root rotation could be modelled by

taking ∆=0.67h. Williams [8] used a beam on an elastic foundation model to analyse an

adhesively-bonded beam specimen. By considering the foundation to be made up of both the

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substrate beam and the adhesive layer, it was shown that the root rotation, ∆, could be

expressed as:

41

41

21

61

+

=

a

a

EE

hh

h∆ (13)

where h and E are the height and Young’s modulus of the substrate respectively, and ha and

Ea are the thickness and Young’s modulus of the adhesive layer respectively. Equation (13)

may be more conveniently expressed as:

41

41

161

+

=

λ∆ Rh (14)

Where λ=2h/ha and R=E/Ea. Clearly, the contribution of the adhesive layer to ∆ depends

upon the values of λ and R. In the present work, ∆ has been obtained via equation (12). The

errors introduced by this simplification were calculated and are discussed in section 5.5.

Thus, by using the root rotation correction as deduced from equation (12), and again making

the simplifying assumption for the geometry factor m, i.e. by using equation (5) rather than

equation (3), ∆, can be simply expressed as:

32

31

364.0 am

⋅

=∆ (15)

Now, replacing a with (a+∆) in equation (10), and substituting for ∆ from equation (15) leads

to:

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−⋅

+= oxa

ma

EbmC

32364.08 3

231

(16)

Finally, differentiating equation (16) and substituting into equation (1) yields the following

expression for GIC:

+=

31

2

2 343.014maEb

mPGIC (17)

In the present work, equations (16) and (17) are described as the corrected beam

theory (CBT) analysis equations for the tapered double cantilever beam specimen. This

analysis has been employed in the new test protocol [9] and has been critically examined

during round-robin testing [4].

2.2.4. Correcting for shear and root rotation via m

As a possible alternative to the new analytical corrections proposed above, the

geometry of the TDCB could be modified to incorporate corrections for root rotation and

shear. The analyses developed in [6,7,10] could be employed to identify a new geometry

factor, m̂ , such that:

( ) constanthh

aha

hham =++=

+=

35.14367.03ˆ23

2

3

2

(18)

Such a modified TDCB specimen could then be defined in which the substrate beams were

profiled such that the value of m̂ was held constant. However, due to the widespread

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popularity of the original TDCB geometry based upon equation (3), the present authors have

considered that the correction scheme outlined in sections 2.2.2-2.2.3 is a better approach and

thus no modification has been made to the beam profile in the present work.

In the following sections, the accuracy of equations (4) and (17) have been assessed by

direct comparison with the Irwin-Kies equation (1) and also for one joint system, with a finite

element analysis employing a cohesive zone model, as will now be described.

3. FINITE ELEMENT (FE) ANALYSIS STUDIES

The tapered double cantilever beam test has been analysed numerically using a

cohesive zone model (CZM) [11]. In the analysis, the variation of cohesive stress as a

function of interfacial opening is defined along the process zone local to the crack tip. The

method has been widely used to predict the global failure by introducing the local fracture

parameters, i.e. the fracture energy GC and a stress parameter, σmax [11-13].

In the present work, cohesive elements based upon a traction-separation law have been

introduced along the crack path at the centre of the adhesive layer assuming a cohesive

fracture, as observed in the experiments. This resulted in a continuum description of the

failure path. A cubic traction-separation law (see Figure 4) was used [14], based on the

adhesive fracture energy GIC and the stress parameter, σmax. The value of σmax was taken to be

equivalent to the uniaxial yield stress of the adhesive, i.e. 50MPa. However, the shape of the

traction-separation is of secondary importance [15]. Once the shape is fixed, the critical

separation, cδ , is no longer an independent parameter. When separation takes place under

both normal and tangential crack face displacements, a potential can be used to generate the

relation between the traction components and the displacements such that the work of

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separation remains equal to GC. The cohesive zone model was incorporated into the finite

element code ABAQUS (version 5.8) via a user subroutine.

4. EXPERIMENTAL STUDIES

4.1. Joint manufacture

Bonded tapered double cantilever beams were manufactured using substrates

consisting of either aluminium alloy (grades ENAW5083 or ENAW2014A) or mild steel (grade

EN32b). Two rubber toughened, structural adhesives were employed; an epoxy-paste

adhesive (ESP110 from Permabond, UK) and an epoxy-film adhesive (AF126 from 3M,

USA). Beams were manufactured using a CNC milling machine to produce a constant

geometry factor, m=2mm-1 as defined in [3] and given by equation (3). The length of the

straight section, xo, was 50mm. All substrates were surface treated such that all crack

propagation occurred cohesively in the adhesive layer. The bond-line thickness of the epoxy-

paste adhesive was controlled during manufacture using 0.4mm diameter wire inserts. When

bonding with the epoxy-film adhesive, a single layer of film was used which produced a final

bond-line thickness of about 0.15mm. The adhesive joints were cured according to the

manufacturers’ instructions. A PTFE insert film of thickness 12.5 microns was inserted into

the adhesive layer at the loading end to create an initial crack. Typically this extended

100mm from the load-line.

4.2. Joint testing

A series of tests were performed following the new protocol [9]. Loading was carried

out at a constant displacement rate of 0.1mm/min. Values of the load, P, the displacement uo

and the crack length, a, were measured for approximately 100mm of crack propagation, i.e.

approximately from x=100 to x=200mm. Full unloading of the joints prior to catastrophic

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failure was always conducted to ensure that the conditions of LEFM were not violated. In all

cases, only elastic deformation of the substrates occurred. The compliance of the tensile

loading system (including all the pins and shackles) was measured by attaching a very stiff

calibration specimen in place of the test specimen and loading to the maximum load attained

in the fracture test. All displacement values recorded from the fracture tests were then

corrected for the system compliance.

5. RESULTS AND DISCUSSION

5.1. Aluminium alloy substrates bonded with the epoxy-paste adhesive

Experimental data from two repeat tests are shown in Figure 5. The values of

compliance, C, have been plotted against crack length, a. When plotting these data, only

values of compliance associated with crack propagation were used, i.e. all crack initiation

points were excluded from the analysis. The experimental data are shown as the unfilled

points. Also shown in Figure 5 are the values of C determined from simple beam theory

(SBT), i.e. from the equation C=8ma/Eb. In addition, the values of C have been determined

using full-profile SBT, i.e. by using equation (10). It should be recalled that full-profile SBT

considers the actual geometry of the beam used, i.e. with the initial non-profiled section, but

does not include the correction for root rotation. The parameters used in the analysis of these

data were:

b= 9.83mm, E=71GPa, ha = 0.4mm, xo=50mm, ho=16mm and m=2mm-1.

From Figure 5, the following observations can be made. Firstly, the C versus a data

from the two experimental tests performed were very repeatable. A single linear regression

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line has been drawn on Figure 5 through these data, but the results of the individual regression

analyses for the two tests respectively are shown in Table I. These data were highly

correlated with a correlation coefficient, r2=0.999 or greater, and an intercept with the crack

length axis at a=17 + 0.6mm. Secondly, the values of C versus a calculated using SBT, i.e.

by assuming C=8ma/Eb, and the values determined using full-profile SBT, i.e. equation (10),

yielded lines with the same slope but different intercepts. These lines intercept the crack

length axis, aint at zero and 33.3mm respectively. The slopes of these analytical lines are

lower than that measured experimentally, i.e. the analytical value of dC/da via SBT or full-

profile SBT is lower than the experimental value as shown in Table I. This observation leads

to values of GIC deduced via equation (4) being significantly lower than values deduced via

equation (1), which uses the experimentally determined values of dC/da. Finally, when

extrapolating the C versus a values back to the C=0 axis as shown in Figure 5, it is important

to remember that all the experimental data were obtained from the profiled section of the

beam and hence the extrapolated lines shown in the figure are a fit to these data.

Figure 6 shows the previous experimental data from Figure 5 but the values of the

compliance predicted by the corrected beam theory (CBT), i.e. equation (16), and also the

values predicted by the finite element cohesive zone model (FE-CZM) are also included. It

should be noted that both of these approaches allow for root rotation effects. It can be seen

that the CBT analysis is a much better fit to the experimental data than was achieved using

either the SBT or full-profile SBT analyses. The results produced by CBT and the FE-CZM

approaches are summarised in Table I. The results from the FE-CZM approach show good

agreement with the CBT analysis.

Figure 7 shows the resistance curve (R-curve) for one test, (Test 1), in which the

values of GIC are plotted against crack length. The values of GIC shown in Figure 7 are all

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associated with crack propagation. The adhesive fracture energy has been calculated using

the: (a) SBT, i.e. equation (4); (b) CBT, i.e. equation (17); and (c) ECM, i.e. equation (1)

approaches. Clearly, the SBT approach is rather conservative, lying 7% below the values

calculated using equation (1). On the other hand, the CBT approach is more accurate (within

2% of the experimental compliance method (ECM)) but is non-conservative in this case (it

should be noted that the GIC axis scale in Figure 7 has been expanded for clarity and does not

start at zero). The above observations demonstrate that the CBT analysis predicts the

compliance of the TDCB specimen more accurately than either the SBT or full-profile SBT,

and leads to an improved accuracy in the values of GIC calculated.

5.2 Aluminium alloy substrates bonded with the epoxy-film adhesive

The aluminium alloy used to manufacture these joints was grade ‘ENAW 2014A.’

The adhesive employed was the single part epoxy-film formulation and a single layer of

adhesive was applied to make the joints. The parameters for the analysis of these joints were:

b= 9.83mm, E=74GPa, ha = 0.15mm, xo=50mm, ho=16mm and m=2mm-1

Tests were performed exactly as before, according to the protocol [9]. The linear regression

analyses performed on the experimental data are summarised in Table II. The values of

dC/da predicted by full-profile SBT and CBT, and the values of aint are also shown in the

table. As before, the regression analysis for SBT has the same slope as full-profile SBT, and

passes through the origin. It can be seen that CBT predicts the experimental value of dC/da

with high accuracy, and hence the agreement between values of GIC calculated via the

experimental compliance method, i.e. equation (1), are in excellent agreement with the values

calculated using the corrected beam theory of equation (17), as shown in Figure 8.

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Figure 8 shows the R-curve for one test, (Test 1), with values of GIC calculated using

the SBT, CBT and ECM analysis methods, (it should be noted that the GIC axis scale in

Figure 8 has been expanded for clarity and does not start at zero). Again, SBT is conservative

and is, on average, 13% below ECM. The CBT analysis however, is just 4% below ECM for

Test 1, and was within 1% for Test 2.

5.3. Mild steel substrates bonded with the epoxy-paste adhesive

These joints were manufactured using mild steel substrates (grade EN32b). The

epoxy-paste adhesive was employed to bond these joints. The parameters for the analysis of

these joints were:

b= 10.0mm, E=207GPa, ha = 0.4mm, xo=50mm, ho=16mm and m=2mm-1

It was apparent from testing these joints that the measured experimental data (i.e. the C versus

a data) were subject to a greater degree of variation than had been observed when using the

aluminium alloy as substrates. The measured data showed greater variation in both the slope

(dC/da) and the intercept (aint) than was observed for the joints manufactured with aluminium

alloy substrates. The mean and standard deviation values of dC/da, aint and correlation

coefficient r2, are shown in Table III for four tests. The values of dC/da and aint predicted by

full-profile SCB and CBT are summarised in the table.

From the results in Table III it is clear that CBT agrees more closely with the

experimental average than full-profile SBT. However, the experimental data were more

scattered, showing comparatively large variations in the linear regression parameters

obtained, as indicated by the standard deviations shown in the table. This observation was

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also noted in the results of the inter-laboratory round-robin programme [5]. The mean

propagation values of GIC from the four tests conducted on the joints consisting of mild steel

substrates bonded with the epoxy-paste adhesive are shown in Table IV, together with the

standard deviations obtained. These values are simply the mean of all the propagation values

recorded via each analysis method.

The reason for the greater degree of scatter noted in these experimental data is likely

to stem from the smaller beam opening displacement values, uo, which are measured when the

stiffer, mild steel substrates were used. The maximum displacement values required to extend

the crack by 100mm was about 2.6mm when mild steel substrates were used and was about

3.6mm when the aluminium alloy substrates were used with the epoxy-paste adhesive.

Although the test data were all corrected for machine compliance effects, the values of uo

were determined from the crosshead travel of the tensile testing machine. Thus, the smaller

displacements occurring during the testing of mild steel joints led to greater errors in the

measured C values and consequently, to greater variation in the experimental values of dC/da

and aint as was shown in Table III. This observation is in line with the trend in the agreement

between the CBT and ECM approaches, namely that joints requiring larger displacement

values to obtain crack propagation show closer agreement between equations (17) and (1).

5.4. A comparison of non-dimensionalised compliance values

It is apparent from equations (10) and (16) that the product CEb, depends only upon

the geometry of the beam, (i.e. upon the geometry factor, m and the distance xo) and on the

crack length, a. Thus, measured values of CEb should be independent of the substrate

material employed. The variation in values of CEb with a have been determined for each

adhesive joint system tested and one example data set for each joint is shown in Figure 9.

Also shown are the values of CEb predicted by the CBT analysis, i.e. by rearranging (16). It

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can be seen that the experimental values of CEb obtained from three different adhesive joint

systems are all in close agreement with the values predicted by the CBT analysis. The joints

consisting of mild steel substrates showed greater variations in the values of CEb, as would be

expected from the earlier observations.

5.5. The accuracy of the root rotation correction

As stated previously, in the present work the root rotation term, ∆, was calculated

using equation (12), which does not consider the adhesive layer in its derivation. Table V

compares the values of ∆, for the three adhesive joint systems employed in the present work,

calculated using equations (12) and (14). Equation (14) accounts for the presence of both the

adhesive layer and the substrate beam. The percentage error in ∆ introduced by ignoring the

adhesive layer is also shown in the table. The values for the Young’s modulus of the

adhesives, Ea, were 4GPa and 2.58GPa for the epoxy-paste and epoxy-film adhesives

respectively. Thus, the values of R, i.e. (E/Ea) for the three adhesive joint systems, in the

order presented were: 17.75, 28.68 and 51.75.

Table V shows the values of ∆ at three different crack lengths approximately

corresponding to the start, middle and end of each test respectively. Clearly, the greatest error

in ∆ is observed for the mild steel substrates, which possess the greatest value of R. The

smallest error in ∆ was observed for the aluminium alloy substrates when bonding the epoxy-

film adhesive. Thus, the use of equation (12) leads to an under correction of beam theory and

it has been calculated [16] that this leads to an error in GIC of about 1% for the joints

manufactured with the aluminium alloy substrates and about 1.7% for joints manufactured

using the mild steel substrates. These relatively small errors demonstrate the acceptability of

- 21 -

neglecting the contribution of the adhesive layer to the root rotation term, ∆, for commonly

employed joint systems.

5.6. The Effect of Residual Stress

In the preceding analysis no account has been taken of residual stress that may arise as

the joints cool from the cure temperature during manufacture. These stresses stem from the

mismatch of thermal expansions between the substrate and the adhesive or from the chemical

shrinkage of the adhesive. The magnitude of the residual stresses introduced also depends

upon the cure temperature of the adhesive. The higher the cure temperature, the greater the

residual stresses become.

Nairn considered the effects of residual stress upon the measured values of GIC for

adhesive joints manufactured using the double cantilever beam geometry [17]. In a more

recent analysis [16] the tapered double cantilever beams considered in the present work were

investigated for residual stress effects. The results demonstrated that, because of the

relatively large values of h employed in the TDCB profile and the modest cure temperatures

that was used to manufacture these joints, the residual stresses caused the calculated values of

GIC to be about 1% too high when the aluminium alloy substrates were employed and about

0.65% too high when the mild steel substrates were used. The residual stress effects were

thus relatively very minor for the results presented in the present work. However, residual

stress effects may become more important when the DCB test geometry is used, where the

value of h is usually lower. Finally, the residual stresses will, of course, change if the

adhesive thickness or adhesive modulus changes, and would usually be greater for adhesives

cured at higher temperatures.

6. CONCLUSIONS

- 22 -

A new corrected beam theory (CBT) analysis, embodied in equations (16) and (17),

has been derived for the tapered double cantilever beam (TDCB) adhesive joint specimen.

This analysis takes into account the discontinuous profile of the beam and proposes a simple

scheme to correct for beam root rotation effects. It has been demonstrated that a previously

published [1-3] simple shear-corrected beam theory analysis leads to errors in the calculated

compliance and also to the calculation of overly conservative values of adhesive fracture

energy, GIC. The new analysis has been compared with results obtained using an

experimental compliance method and also using a finite element analysis approach employing

a cohesive zone model. Close agreement in the results from these two approaches and the

new CBT analysis was observed. The various simplifications in the new analysis have been

shown to introduce only small errors when compared to the errors incurred by neglecting the

effects of beam root rotation and by not considering the real profile of the beam. This new

corrected beam theory approach is now embodied in the new British Standard [9].

Acknowledgement---The authors wish to thank the EPSRC (Advanced Fellowship

AF/992781) and the National Physical Laboratory for financial support. For performing the

residual stress calculations, the authors wish to thank Professor John Nairn at the University

of Utah. For providing a valuable discussion forum, the authors wish to thank the European

Structural Integrity Society’s Technical Committee (4) on Polymers, Adhesives and

Composites.

REFERENCES

1. Ripling, E.J., S. Mostovoy, and R.L. Patrick, Measuring fracture toughness of

adhesive joints. Materials Research & Standards (ASTM Bulletin), 1964. 4(3, March):

p. 129-134.

- 23 -

2. Mostovoy, S., P.B. Crosley, and E.J. Ripling, Use of crack-line loaded specimens for

measuring plane-strain fracture toughness. Journal of Materials, 1967. 2(3): p. 661-

681.

3. ASTM, ASTM D3433, in Annual book of ASTM standards. Adhesives section 15.

1990: Philadelphia.

4. Blackman, B.R.K. and A.J. Kinloch, Fracture tests on structural adhesive joints, in

Fracture mechanics testing methods for polymers, adhesives and composites, D.R.

Moore, A. Pavan, and J.G. Williams, Editors. 2001, Elsevier Science Ltd.:

Amsterdam. p. 225-267.

5. Blackman, B.R.K. and A.J. Kinloch, Measuring the Mode I adhesive fracture energy

of structural adhesive joints: The results of a European round-robin. to be published,

2001.

6. Kanninen, M.F., An augmented double cantilever beam model for studying crack

propagation and arrest. International Journal of Fracture, 1973. 9(1): p. 83-92.

7. Kanninen, M.F., A dynamic analysis of unstable crack propagation and arrest in the

DCB test specimen. International Journal of Fracture, 1974. 10(3): p. 415-431.

8. Williams, J.G. Fracture in adhesive joints: The beam on elastic foundation model. in

Proc. International Mechanical Engineering Congress & Exhibition. ASME

Symposium on Mechanics of Plastics and Plastics Composites. 12-17 November 1995.

San Francisco, USA.

9. Blackman, B.R.K. and A.J. Kinloch, Determination of the mode I adhesive fracture

energy, GIC, of structural adhesives using the double cantilever beam (DCB) and the

tapered double cantilever beam (TDCB) specimens. ESIS TC4 Protocol, 2000 and

British Standard BS 7991-2001.

10. Williams, J.G., End corrections for orthotropic DCB specimens. Composites Science

and Technology, 1989. 35: p. 367-376.

- 24 -

11. Tvergaard, V. and J.W. Hutchinson, The relation between crack growth resistance and

fracture process parameters in elastic-plastic solids. Journal of Mechanics and

Physics of Solids, 1992. 40: p. 1377-1397.

12. Tvergaard, V. and J.W. Hutchinson, The influence of plasticity on mixed mode

interface toughness. Journal of Mechanics and Physics of Solids, 1993. 41: p. 1119-

1135.

13. Needleman, A.A., A continuum model for void nucleation by inclusion debonding.

Journal of Applied Mechanics, 1987. 54: p. 525-531.

14. Busso, E., Chen, J., Crisfield, M., Kinloch, A.J., Matthews, F.L., and Qui, Y.,

Predicting progressive delamination of composite material specimens via interface

elements. Mechanics of Composite Materials and Structures, 1999. 6, p301-318.

15. Williams, J.G. and H. Hadavinia, Analytical solution of cohesive zone models. Journal

of Mechanics and Physics of Solids (in press), 2001.

16. Nairn, J.A., Personal communication. 2000.

17. Nairn, J.A., Energy release rate analysis for adhesive and laminate double cantilever

beam specimens emphasizing the effect of residual stresses. International Journal of

Adhesion and Adhesives, 1999. 20: p. 59-70.

- 25 -

Table I. Results of the linear regression analyses for aluminium alloy substrates bonded with

the epoxy-paste adhesive.

aint (mm) dC/da (N-1) r2

Test 1 17.6 2.46 x 10-5 0.9990 a

Test 2 16.5 2.47 x 10-5 0.9997 a

SBT: (C=8ma/Eb) 0 2.29 x 10-5 1.0

full-profile SBT: eqn. (10) 33.3 2.29 x 10-5 1.0

CBT: eqn. (16) 26.5 2.50 x 10-5 1.0

FE-CZM 20.1 2.38 x 10-5 1.0

(a) Initiation values from the tests are not included in the regression analyses.

- 26 -

Table II. Results of the linear regression analysis for aluminium alloy substrates bonded with

the epoxy-film adhesive.

aint (mm) dC/da (N-1) r2

Test 1 25.8 2.61 x 10-5 0.9986 a

Test 2 21.6 2.38 x 10-5 0.9965 a

full-profile SBT: eqn. (10) 33.3 2.17x 10-5 1.0

CBT: eqn. (16) 26.5 2.38x 10-5 1.0

(a) Initiation values from the tests are not included in the regression analyses.

- 27 -

Table III. Results of the linear regression analysis for mild steel substrates bonded with the

epoxy-paste adhesive.

aint (mm) dC/da (N-1) r2

Experimental b 10.0 +15.3 8.77+1.22 x 10-6 0.9886+0.01 a

full-profile SBT: eqn. (10) 33.3 7.73 x 10-6 1.0

CBT: eqn. (16) 26.5 8.51 x 10-6 1.0

(a) Initiation values from the tests are not included in the regression analyses. (b) The (+)

indicate the standard deviations obtained from four repeat tests.

- 28 -

TABLE IV. Values of GIC obtained from four repeat tests on joints consisting of mild steel

substrates bonded with the epoxy-paste adhesive.

GIC (J/m2)

Analysis method SBT (eqn. 4) CBT (eqn. 17) ECM (eqn. 1)

Mean value 930 1019 1060

Standard deviation 56 62 190

- 29 -

Table V. The root rotation term ∆ calculated using equations (12) and (14) for the adhesive

joints used in the present study, and the % error in ∆.

Adhesive joint λ ∆ eqn. (12) ∆ eqn. (14) % Error

(For a=101mm, h=24.99mm)

Al-alloy/epoxy-paste 119.02 15.97 16.53 3.42

Al-alloy/epoxy-film 333.25 15.97 16.30 2.04

Mild steel/epoxy-paste 119.02 15.97 17.48 8.63

(For a=150mm, h=32.48mm)

Al-alloy/epoxy-paste 154.79 20.76 21.33 2.68

Al-alloy/epoxy-film 433.12 20.76 21.09 1.59

Mild steel/epoxy-paste 154.69 20.76 22.31 6.96

(For a=200mm, h=39.32mm)

Al-alloy/epoxy-paste 187.22 25.12 25.70 2.24

Al-alloy/epoxy-film 524.21 25.12 25.46 1.32

Mild steel/epoxy-paste 187.22 25.12 26.70 5.92

- 30 -

Loading holes

CrackAdhesive

Substrate

Substrate

Figure 1. The tapered double cantilever beam (TDCB) adhesive joint test specimen.

- 31 -

Pb

hoh

xxo

Figure 2. Single arm of the TDCB test specimen without adhesive, showing notation and

loading.

- 32 -

0

0.001

0.002

0.003

0.004

0.005

0.006

0 50 100 150 200 250 300

C (m

m/N

)

x (mm)

C α x3

C α x

(2/3) xo

DCBsection

TDCBsection

Figure 3. The variation in compliance, C, with crack position, x, for a tapered double

cantilever beam specimen with xo=50mm, ho=16mm, E=71GPa, m=2mm-1 and b=9.81mm.

The initial straight part of the beam is denoted as the DCB section, and the height profiled

part is denoted as the TDCB section. (Solid line is equation (10), the dashed line is the

extrapolation of the linear TDCB section back to C=0.)

- 33 -

Figure 4. The cubic traction-separation law employed in the cohesive zone model analysis.

- 34 -

0

0.001

0.002

0.003

0.004

0.005

0 50 100 150 200 250

Test 1Test 2Full-profile SBT: eqn. (10)SBT: 8ma/Eb

C (m

m/N

)

Crack length (mm)

Figure 5. The variation of compliance with crack length for a tapered double cantilever beam

specimen manufactured with aluminium alloy substrates and bonded with the epoxy-paste

adhesive. Experimental results from two tests are shown, together with the values predicted

by the full-profile SBT, i.e. equation (10), and from SBT, i.e. C=8ma/Eb. The solid line is a

regression line through the experimental data.

- 35 -

0

0.001

0.002

0.003

0.004

0.005

0 50 100 150 200 250

Test 1Test 2CBT eqn. (16)FEA-CZM

C (m

m/N

)

Crack length (mm)

Figure 6. The variation of compliance with crack length for a tapered double cantilever beam

specimen manufactured with aluminium alloy substrates and bonded with the epoxy-paste

adhesive. Experimental results from two tests are shown, together with the values predicted

by CBT, i.e. equation (16), and by using the finite element cohesive zone model (FE-CZM).

- 36 -

500

550

600

650

700

750

800

100 120 140 160 180 200

SBT, eqn (4)CBT, eqn (17)ECM, eqn (1)

GIC

(J/m

2 )

Crack length (mm)

Figure 7. R-curve behaviour for a tapered double cantilever beam specimen manufactured

with aluminium alloy substrates and bonded with the epoxy-paste adhesive. Values of GIC

were deduced using; simple beam theory (SBT) i.e. equation (4), corrected beam theory

(CBT), i.e. equation (17), and the experimental compliance method (ECM), i.e. equation (1).

- 37 -

600

800

1000

1200

1400

1600

1800

80 100 120 140 160 180

SBT eqn. (4)CBT eqn. (17)ECM eqn. (1)

GIC

(J/m

2 )

Crack length (mm)

Figure 8. R-curve behaviour for a tapered double cantilever beam specimen manufactured

with aluminium alloy substrates and bonded with the epoxy-film adhesive. Values of GIC

were deduced using; simple beam theory (SBT), i.e. equation (4), corrected beam theory

(CBT), i.e. equation (17), and the experimental compliance method (ECM), i.e. equation (1).

- 38 -

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250

CBT eqn. (16)

Al-alloy/epoxy-paste

Al-alloy/epoxy-film

steel/epoxy-paste

CE

b (d

imen

sion

less

)

Crack length (mm)

Figure 9. The variation in the values of CEb with crack length for the three adhesive joint

systems investigated in the present work. Experimental values are compared to the values

predicted by the corrected beam theory, i.e. equation (16).